Ponder This

July 2004

This month's puzzle comes from Aditya K Prasad, who heard it from a friend. It is similar to our November 2002 puzzle (from Martin Gardner's February 1979 Scientific American column), but with an important difference. To be considered for publication, please supply answers for BOTH parts of the puzzle.

Part 1:
There is a round table divided into 4 equal quadrants, with one cup in each quadrant. The quadrants are labeled with letters (A, B, C, D) that do not move. Initially, each cup is randomly face-up or face-down. You are blindfolded and put in front of the table. On each turn of the game, you instruct a genie to flip the cups in whichever positions you choose (e.g., you may say "flip the cups in A and B"), possibly choosing no cups or possibly all four. The genie complies. At this point, if all the cups on the table are face-up, the genie will tell you that you have won the game and are free to go. If not, he rotates the cups randomly (possibly not rotating them) and you play another turn. Give a strategy to win this game in a finite number of moves (the solution is not unique).

Remark: the outcome of "rotation" the four cups is one of the four possible positions: the cups originally at (A,B,C,D) can be at (A,B,C,D), (B,C,D,A), (C,D,A,B), or (D,A,B,C). It is not an arbitrary permutation.

Notice that you cannot examine the current orientation of any cup at any time. This contrasts with the earlier puzzle.

Part 2:
Suppose that instead of 4, there are n divisions and cups. For which n is it possible to guarantee a win? Prove your answer is correct.

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